Register to reply

Is there a way of approximating e^-x for large x

by Chain
Tags: approximating
Share this thread:
Chain
#1
Dec22-12, 09:38 AM
P: 28
I'm trying to evaluate an integral with e^-x where x is huge in the domain of the integral so I can't evaluate it numerically without making an approximation.
Phys.Org News Partner Science news on Phys.org
Security CTO to detail Android Fake ID flaw at Black Hat
Huge waves measured for first time in Arctic Ocean
Mysterious molecules in space
mfb
#2
Dec22-12, 10:01 AM
Mentor
P: 11,602
Can you add more context? There might be a clever way to approximate e^(-x), but if that value is not added to something, e^(-x)=e^(-x-x0)*e^(x0) where the second factor is independent of x and the first factor can be chosen to be about e^0.
Chain
#3
Dec22-12, 10:15 AM
P: 28
The integral is 4*(r^2)*exp(-2*r/a)/a^3 integrated between 1 and ∞ (The probability an electron in the ground state of hydrogen is more than 1 metre away from the nucleus) a=0.529*10^-10

Hurkyl
#4
Dec22-12, 11:14 AM
Emeritus
Sci Advisor
PF Gold
Hurkyl's Avatar
P: 16,092
Is there a way of approximating e^-x for large x

Note that your integrand is one you can anti-differentiate without much trouble, so you can get the exact answer. (e.g. integration by parts. Or computer algebra package)
mfb
#5
Dec22-12, 12:17 PM
Mentor
P: 11,602
If you just want an estimate: The exponential will drop very quick, so regions with r>1m+eps are irrelevant and r^2 is nearly constant and =1m^2. Therefore, the integral is simply 4m^2/a^3 * exp(-2r/a) which can be evaluated as 2m^2/a^2 * exp(-2m/a) ≈ 8*10^20 * exp(-4*10^10) ≈ 10^(-10^10) where the last approximation is very rough.
mathman
#6
Dec22-12, 02:39 PM
Sci Advisor
P: 6,039
e-x = 10-0.43429448190325x. The constant is log10e.

Using the above for large x, you can separate the integer and fractional parts of the exponent. I assume you know how to proceed from here.
lurflurf
#7
Dec22-12, 09:46 PM
HW Helper
P: 2,263
Are you talking about Gauss–Laguerre quadrature?

[tex]\int_0^\infty f(x) e^{-x} \mathop{\text{dx}}\sim \sum_{i=1}^n w_i f(x_i)[/tex]

where xi are zeros of a Laguerre polynomial and

[tex]w_i=\frac{x_i}{(n+1)^2[L_{n+1}(x_i)]^2}[/tex]
Chain
#8
Dec23-12, 02:23 AM
P: 28
Fair enough, yeah I realised after posting this the integral could be solved analytically >__< and I got a valule of something like 10^(-10^10) but thanks for the responses :)


Register to reply

Related Discussions
Approximating x*log(x) as x->0 Calculus 0
Approximating PI Calculus & Beyond Homework 4
Approximating Integrals Calculus 0
Approximating PI General Math 1
Solving large #s raised to a large # with the mod function Precalculus Mathematics Homework 1